Breuckmann, NP; Eberhardt, JN; (2021) Balanced Product Quantum Codes. IEEE Transactions on Information Theory 2021 , 67 (10) pp. 6653-6674. 10.1109/TIT.2021.3097347.

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Abstract

This work provides the first explicit and non-random family of [[N,K,D]] LDPC quantum codes which encode K∈Θ(N45) logical qubits with distance D∈Ω(N35) . The family is constructed by amalgamating classical codes and Ramanujan graphs via an operation called balanced product . Recently, Hastings–Haah–O’Donnell and Panteleev–Kalachev were the first to show that there exist families of LDPC quantum codes which break the polylog(N)N−−√ distance barrier. However, their constructions are based on probabilistic arguments which only guarantee the code parameters with high probability whereas our bounds hold unconditionally. Further, balanced products allow for non-abelian twisting of the check matrices, leading to a construction of LDPC quantum codes that can be shown to have K∈Θ(N) and that we conjecture to have linear distance D∈Θ(N) .

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